Peristaltic pumps in one known form comprise a cylindrical housing for a flexible resilient tube entering at one point on the circumference thereof and extending around the interior of the housing for approximately 260.degree. to exit at another point on the housing substantially confronting the entry point. A roller is forced into compressive engagement with the tube and rotated by gear means on the axis of the housing to provide the means for drawing a fluid substance in through one end of the tube and forcing it around the tube to the outlet by progressively compressively collapsing the tube. By the provision of a restriction at the outlet side of the pump and the rotation of the roller at a relatively high speed the fluid substance can be pumped out of the tube at a higher pressure than that derived from its source. In another known form the tube has been positioned within the housing to circumscribe a complete helix with its inlet and outlet end portions extending substantially tangentially with respect to the cylindrical housing. Both forms of pumps have not been satisfactory for the many applications requiring substantially uniform flow of fluid, i.e. as opposed to a pulsating flow. It can be readily appreciated that such decompressurization of the fluid substance can in fact induce reverse flow thus requiring a non-return or check valve to be installed to avoid such unwarranted and dangerous effect if the pump is being utilized for surgical purposes, such as blood transfusion.